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In mathematics a cocycle is a closed cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likewise used in group cohomology. In autonomous dynamical systems, cocycles are used to describe particular kinds of map, as in Oseledets theorem.




Definition






= Algebraic Topology

=
Let X be a CW complex and




C

n


(
X
)


{\displaystyle C^{n}(X)}

be the singular cochains with coboundary map




d

n


:

C

n
−
1


(
X
)
→

C

n


(
X
)


{\displaystyle d^{n}:C^{n-1}(X)\to C^{n}(X)}

. Then elements of




ker

d


{\displaystyle {\text{ker }}d}

are cocycles. Elements of




im

d


{\displaystyle {\text{im }}d}

are coboundaries. If



φ


{\displaystyle \varphi }

is a cocycle, then



d
∘
φ
=
φ
∘
∂
=
0


{\displaystyle d\circ \varphi =\varphi \circ \partial =0}

, which means cocycles vanish on boundaries.




See also


Čech cohomology
Cocycle condition




References

Kata Kunci Pencarian:

• Cocycle
• Čech cohomology
• Cocycle category
• Oseledets theorem
• Teichmüller cocycle
• Kodaira–Spencer map
• Factor system
• Tomita–Takesaki theory
• Loop algebra
• Cup product

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